# Tagged Questions

Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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134 views

### Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
0answers
105 views

### Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
1answer
249 views

### Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
1answer
50 views

### Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
0answers
50 views

### Generalized fundamental theorem of Galois theory

In the generalized fundamental theorem of Galois theory, theorem says there is a one-to-one correspondence between the set of all intermediate fields of extension and the set of all $\textbf{closed}$ ...
0answers
69 views

### A Problem about Galois Extension

This is a basic question about Galois Extension, but I want some details about it. Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield ...
3answers
101 views

### Automorphisms on a field $F$

I am trying to understand this proposition with respect to algebraic closures of a field $F$ Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure ...
0answers
204 views

### Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
1answer
74 views

### composition of a $\mathbb{Z}_p$-extensions and Hilbert class field extension (Iwasawa)

I am reading Iwasawa's article On $\Gamma$-extensions of algebraic number fields. In paragraph 7.3 : " We now take as M the maximal unramified abelian $p$-extension of $L$ in $\Omega$ , i.e. the ...
0answers
50 views

### question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
1answer
49 views

### Uniqueness of $p^{th}$ powers in characteristic $p$

The other day, my undergrad Galois theory professor used the fact that in char $p$, $p$th powers exist and are unique. How can one understand why uniqueness holds? Thanks
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59 views

### $\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism ...
3answers
435 views

### Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
2answers
162 views

### The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
2answers
223 views

1answer
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### Galois group of a quartic

Let $x^4+ax^2+b$ in $K[x]$ (with char $K\neq$2) be irreducible with Galois group $G$. (a) If $b$ is a square in $K$, then $G = \mathbb{Z}_2\times\mathbb{Z}_2$. (b) If $b$ is not a square in $K$ ...
2answers
103 views

### Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
1answer
48 views

1answer
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### My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
1answer
123 views

### Is this proof that $\mathbb F_q^*$ is cyclic correct?

I would like to show that the multiplicative group of a finite field is cyclic. My goal is to use the best of my knowledge to render the argument as efficient yet simple and understandable as ...